Fundamentals of Psychoacoustics
171
roughness
masker
C.4
Critical Band
As illustrated in figure 6 of chapter 2, two pure tones whose frequencies are close
to each other give rise to the phenomenon of beating. In formula, from simple
trigonometry
sin
1
t + sin
2
t = 2 sin
(
1
+
2
)t
2
cos
(
1
-
2
)t
2
,
(16)
where the first sinusoidal term in the product can be interpreted as a carrier
signal modulated by the second, cosinusoidal term.
As we vary the distance between the frequencies
1
and
2
, the resulting
sound is perceived differently, and a sense of roughness emerges for distances
smaller than a certain threshold. A schematic view of the sensed signal is rep-
resented in figure 7. The solid lines may be interpreted as time-varying sensed
pitch tracks. If they are far enough we perceive two tones. When they get closer,
at a certain point a sensation of roughness emerges, but they are still resolved.
As they get even closer, we stop perceiving two separate tones and, at a certain
point, we hear a single tone that beats. Also, when they are very close to each
other, the roughness sensation decreases.
Roughness
(Critical Band)
One Tone
Beats
t
t
1
2
Figure 7: Schematic representation of the subjective phenomena of beats and
roughness (adapted from [86])
The region where roughness gets in defines a critical band, and that fre-
quency region roughly corresponds to the segment of basilar membrane that
gets excited by the tone at frequency
1
. The sensation of roughness is related
with that property of sound quality that is called consonance, and that can
be evaluated along a continuous scale, as reported in figure 8. We notice that
the maximum degree of dissonance is found at about one quarter of critical
bandwidth.
C.5
Masking
When a sinusoidal tone impinges the outer ear, it propagates mechanically until
the basilar membrane, where it affects the reception of other sinusoidal tones
at nearby frequencies. If the incoming 400Hz tone, called the masker, has 70dB
of IL, a tone at 600Hz has to be more than 30dB louder than its miniminal